Existence of solutions in the sense of distributions of anisotropic nonlinear elliptic equations with variable exponent

The aim of this paper is to study the existence of solutions in the sense of distributions for a~strongly nonlinear elliptic problem where the second term of the equation $f$ is in $W^{-1,\overrightarrow{p}'(\,\cdot\,)}(\Omega)$ which is the dual space of the anisotropic Sobolev $W_{0}^{1,\overrightarrow{p}(\,\cdot\,)}(\Omega)$ and later $f$ will be in~$L^{1}(\Omega)$

Strongly nonlinear nonhomogeneous elliptic unilateral problems with L^1 data and no sign conditions

In this article, we prove the existence of solutions to unilateral problems involving nonlinear operators of the form: $$Au+H(x,u,abla u)=f$$ where $A$ is a Leray Lions operator from $W_0^{1,p(x)}(Omega)$ into its dual $W^{-1,p'(x)}(Omega)$ and $H(x,s,xi)$ is the nonlinear term satisfying some growth condition but no sign condition. The right hand side $f$ belong to $L^1(Omega)$